Group #34079846 — subgroups

Subgroups — #34079846

Order 19328.

446 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752760	C₂	1	2	yes	maximal, complemented
2	#23752610	C₂	1	1	yes	maximal, characteristic, complemented
3	#23752558	C₂	1	2	yes	maximal, complemented
4	#23752462	C₂	1	4	yes	maximal, complemented
5	#23752324	C₂	1	4	yes	maximal, complemented
6	C₄×#14558987	C₂	1	1	yes	maximal, characteristic, complemented
7	#493×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14559003	—	2	2	no	—
9	C₂×#14558987	C₂²	1	2	yes	—
10	#18012739	C₂²	1	2	yes	—
11	#18012726	—	2	2	no	—
12	#18012710	—	2	2	no	—
13	#18012710	C₂²	1	4	yes	—
14	D₄×Dic₁₅₁	—	2	2	no	—
15	D₄×Dic₁₅₁	C₂²	1	4	yes	—
16	C₄²×D₁₅₁	C₂²	1	1	yes	characteristic
17	#18012671	C₂²	1	4	yes	—
18	#18012660	C₂²	1	8	yes	—
19	#18012649	C₂²	1	2	yes	—
20	C₄×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×D₈×C₁₅₁	C₂²	1	4	yes	—
22	#127×C₁₅₁	C₂²	1	2	yes	—
23	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
24	#14559003	—	4	8	no	—
25	C₂×Dic₃₀₂	—	2	2	no	—
26	C₂×C₂²⋊D₁₅₁	—	2	4	no	—
27	C₂²×Dic₁₅₁	—	2	4	no	—
28	C₂×C₄×D₁₅₁	D₄	1	2	yes	—
29	C₂×C₄×D₁₅₁	C₂³	1	1	yes	characteristic
30	#14558987	—	2	4	no	—
31	C₂×C₁₅₁⋊C₈	C₂³	1	2	yes	—
32	C₄⋊Dic₁₅₁	—	2	2	no	—
33	C₄⋊Dic₁₅₁	C₂³	1	4	yes	—
34	C₂²⋊Dic₁₅₁	—	2	4	no	—
35	C₂²⋊Dic₁₅₁	—	2	4	no	—
36	C₄×Dic₁₅₁	D₄	1	2	yes	—
37	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
38	C₂×D₄×C₁₅₁	—	2	2	no	—
39	C₂×D₄×C₁₅₁	—	2	2	no	—
40	C₂×D₄×C₁₅₁	C₂³	1	4	yes	—
41	D₈×C₁₅₁	—	2	8	no	—
42	C₂×C₈×C₁₅₁	C₂³	1	2	yes	—
43	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
44	C₂²×D₁₅₁	C₂×D₄	1	1	yes	characteristic
45	Dic₃₀₂	—	4	4	no	—
46	C₂²⋊D₁₅₁	—	4	8	no	—
47	C₂×Dic₁₅₁	—	2	4	no	—
48	C₂×Dic₁₅₁	—	4	8	no	—
49	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₁₅₁	C₂×D₄	1	2	yes	—
51	C₄×D₁₅₁	—	2	4	no	—
52	C₄×D₁₅₁	—	2	2	no	—
53	C₁₅₁⋊C₈	—	2	2	no	—
54	C₂³×C₁₅₁	—	2	4	no	—
55	D₄×C₁₅₁	—	2	8	no	—
56	D₄×C₁₅₁	—	4	4	no	—
57	D₄×C₁₅₁	—	4	4	no	—
58	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
60	C₈×C₁₅₁	#41	1	4	yes	complemented
61	C₂²×C₁₅₁	—	4	4	no	—
62	C₂²×C₁₅₁	—	4	4	no	—
63	C₂²×C₁₅₁	—	4	4	no	—
64	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₁₅₁	—	2	1	no	—
66	C₂×D₁₅₁	—	2	2	no	—
67	C₄×C₁₅₁	AGL(1,ℤ/8)	1	4	yes	—
68	C₄×C₁₅₁	C₂×#41	1	2	yes	—
69	Dic₁₅₁	—	2	1	no	—
70	Dic₁₅₁	—	2	2	no	—
71	Dic₁₅₁	—	4	4	no	—
72	C₂×C₁₅₁	—	8	4	no	—
73	C₂×C₁₅₁	#532	1	1	yes	characteristic
74	C₂×C₁₅₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
75	D₁₅₁	—	4	1	no	—
76	C₁₅₁	#3184	1	1	yes	characteristic, complemented, Sylow
77	#3184	—	151	1	no	maximal, Sylow
78	#532	—	151	2	no	—
79	#498	—	151	1	no	—
80	#493	D₁₅₁	1	1	yes	characteristic, complemented
81	#486	—	151	2	no	—
82	#469	—	151	4	no	—
83	#442	—	151	4	no	—
84	C₄×M₁₆	—	151	1	no	—
85	C₂×#41	—	302	2	no	—
86	C₂×D₈	C₂×D₁₅₁	1	4	yes	—
87	C₂×M₁₆	—	151	2	no	—
88	C₄⋊Q₈	—	151	2	no	—
89	#127	C₂×D₁₅₁	1	2	yes	—
90	#124	—	302	2	no	—
91	#121	—	302	2	no	—
92	#121	—	151	4	no	—
93	C₄×D₄	—	302	2	no	—
94	C₄×D₄	—	151	4	no	—
95	C₂×C₄²	—	151	1	no	—
96	#106	—	151	4	no	—
97	#102	—	151	8	no	—
98	#97	—	151	2	no	—
99	C₄×C₈	—	151	1	no	—
100	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
101	#41	—	604	8	no	—
102	C₂×Q₈	—	302	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	—	302	4	no	—
106	C₂×D₄	C₂²×D₁₅₁	1	4	yes	—
107	C₂²×C₄	—	302	4	no	—
108	C₂²×C₄	—	151	2	no	—
109	C₂²×C₄	—	151	1	no	—
110	D₈	—	2	8	no	—
111	M₁₆	—	302	4	no	—
112	C₂×C₈	—	151	2	no	—
113	C₂×C₈	C₂²×D₁₅₁	1	2	yes	—
114	C₄⋊C₄	—	302	2	no	—
115	C₄⋊C₄	—	151	4	no	—
116	C₂²⋊C₄	—	302	4	no	—
117	C₂²⋊C₄	—	302	4	no	—
118	C₄²	—	151	2	no	—
119	C₄²	—	151	1	no	—
120	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
121	C₂³	—	2	4	no	—
122	C₂³	—	151	1	no	—
123	Q₈	—	604	4	no	—
124	D₄	—	2	8	no	—
125	D₄	—	4	4	no	—
126	D₄	—	4	4	no	—
127	D₄	—	604	8	no	—
128	C₂×C₄	—	302	4	no	—
129	C₂×C₄	—	302	2	no	—
130	C₂×C₄	—	302	4	no	—
131	C₂×C₄	—	604	8	no	—
132	C₂×C₄	—	151	1	no	—
133	C₂×C₄	—	151	2	no	—
134	C₂×C₄	D₄×D₁₅₁	1	2	yes	—
135	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
136	C₈	—	302	2	no	—
137	C₈	#14559003	1	4	yes	complemented
138	C₂²	—	4	4	no	—
139	C₂²	—	4	4	no	—
140	C₂²	—	4	4	no	—
141	C₂²	—	302	2	no	—
142	C₂²	—	302	1	no	—
143	C₂²	C₂×D₄×D₁₅₁	1	1	yes	characteristic
144	C₄	—	302	2	no	—
145	C₄	—	302	1	no	—
146	C₄	—	604	4	no	—
147	C₄	#18012763	1	4	yes	—
148	C₄	C₂×#14559003	1	2	yes	—
149	C₂	—	8	4	no	—
150	C₂	—	604	1	no	—
151	C₂	#23752760	1	1	yes	characteristic
152	C₂	C₂×#18012763	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23752760

C₂

yes

maximal, complemented

#23752610

C₂

yes

maximal, characteristic, complemented

#23752558

C₂

yes

maximal, complemented

#23752462

C₂

yes

maximal, complemented

#23752324

C₂

yes

maximal, complemented

C₄×#14558987

C₂

yes

maximal, characteristic, complemented

#493×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#14559003

—

C₂×#14558987

C₂²

yes

—

#18012739

C₂²

yes

—

#18012726

—

#18012710

—

#18012710

C₂²

yes

—

D₄×Dic₁₅₁

—

D₄×Dic₁₅₁

C₂²

yes

—

C₄²×D₁₅₁

C₂²

yes

characteristic

#18012671

C₂²

yes

—

#18012660

C₂²

yes

—

#18012649

C₂²

yes

—

C₄×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₂×D₈×C₁₅₁

C₂²

yes

—

#127×C₁₅₁

C₂²

yes

—

C₄×C₈×C₁₅₁

C₂²

yes

characteristic

#14559003

—

C₂×Dic₃₀₂

—

C₂×C₂²⋊D₁₅₁

—

C₂²×Dic₁₅₁

—

C₂×C₄×D₁₅₁

D₄

yes

—

C₂×C₄×D₁₅₁

C₂³

yes

characteristic

#14558987

—

C₂×C₁₅₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

C₂³

yes

—

C₂²⋊Dic₁₅₁

—

C₂²⋊Dic₁₅₁

—

C₄×Dic₁₅₁

D₄

yes

—

C₄×Dic₁₅₁

C₂³

yes

characteristic

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

C₂³

yes

—

D₈×C₁₅₁

—

C₂×C₈×C₁₅₁

C₂³

yes

—

C₄²×C₁₅₁

C₂³

yes

characteristic

C₂²×D₁₅₁

C₂×D₄

yes

characteristic

Dic₃₀₂

—

C₂²⋊D₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₂×D₄

yes

characteristic

C₂×Dic₁₅₁

C₂×D₄

yes

—

C₄×D₁₅₁

—

C₄×D₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂⁴

yes

characteristic

C₈×C₁₅₁

#41

yes

complemented

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂²×D₄

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₄×C₁₅₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₅₁

C₂×#41

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#532

yes

characteristic

C₂×C₁₅₁

C₂×AGL(1,ℤ/8)

yes

—

D₁₅₁

—

C₁₅₁

#3184

yes

characteristic, complemented, Sylow

#3184

—

151

maximal, Sylow

#532

—

151

—

#498

—

151

—

#493

D₁₅₁

yes

characteristic, complemented

#486

—

151

—

#469

—

151

—

#442

—

151

—

C₄×M₁₆

—

151

—

C₂×#41

—

302

—

C₂×D₈

C₂×D₁₅₁

yes

—

C₂×M₁₆

—

151

—

C₄⋊Q₈

—

151

—

#127

C₂×D₁₅₁

yes

—

#124

—

302

—

#121

—

302

—

#121

—

151

—

C₄×D₄

—

302

—

C₄×D₄

—

151

—

C₂×C₄²

—

151

—

#106

—

151

—

#102

—

151

—

#97

—

151

—

C₄×C₈

—

151

—

100

C₄×C₈

C₂×D₁₅₁

yes

characteristic

101

#41

—

604

—

102

C₂×Q₈

—

302

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

302

—

106

C₂×D₄

C₂²×D₁₅₁

yes

—

107

C₂²×C₄

—

302

—

108

C₂²×C₄

—

151

—

109

C₂²×C₄

—

151

—

110

D₈

—

111

M₁₆

—

302

—

112

C₂×C₈

—

151

—

113

C₂×C₈

C₂²×D₁₅₁

yes

—

114

C₄⋊C₄

—

302

—

115

C₄⋊C₄

—

151

—

116

C₂²⋊C₄

—

302

—

117

C₂²⋊C₄

—

302

—

118

C₄²

—

151

—

119

C₄²

—

151

—

120

C₄²

C₂²×D₁₅₁

yes

characteristic

121

C₂³

—

122

C₂³

—

151

—

123

Q₈

—

604

—

124

D₄

—

125

D₄

—

126

D₄

—

127

D₄

—

604

—

128

C₂×C₄

—

302

—

129

C₂×C₄

—

302

—

130

C₂×C₄

—

302

—

131

C₂×C₄

—

604

—

132

C₂×C₄

—

151

—

133

C₂×C₄

—

151

—

134

C₂×C₄

D₄×D₁₅₁

yes

—

135

C₂×C₄

C₂³×D₁₅₁

yes

characteristic

136

C₈

—

302

—

137

C₈

#14559003

yes

complemented

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

302

—

142

C₂²

—

302

—

143

C₂²

C₂×D₄×D₁₅₁

yes

characteristic

144

C₄

—

302

—

145

C₄

—

302

—

146

C₄

—

604

—

147

C₄

#18012763

yes

—

148

C₄

C₂×#14559003

yes

—

149

C₂

—

150

C₂

—

604

—

151

C₂

#23752760

yes

characteristic

152

C₂

C₂×#18012763

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#493×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
hypercenter	#493	D₁₅₁
hypocenter	C₁₅₁	#3184
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

C₂×D₄×D₁₅₁

commutator, cosocle

C₂×C₄×C₁₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₁

Fitting

#493×C₁₅₁

C₂

socle

C₂²×C₁₅₁

C₂²×D₄

D₁₅₁

C₁₅₁

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #23752760	C₂	semidirect
2. #23752610	C₂	semidirect
3. #23752558	C₂	semidirect
4. #23752462	C₂	semidirect
5. #23752324	C₂	semidirect
6. C₄×#14558987	C₂	semidirect
7. #493×C₁₅₁	C₂	semidirect
60. C₈×C₁₅₁	#41	semidirect
76. C₁₅₁	#3184	semidirect
80. #493	D₁₅₁	semidirect
137. C₈	#14559003	semidirect

H ≅ G/N

type

1. #23752760

C₂

semidirect

2. #23752610

C₂

semidirect

3. #23752558

C₂

semidirect

4. #23752462

C₂

semidirect

5. #23752324

C₂

semidirect

6. C₄×#14558987

C₂

semidirect

7. #493×C₁₅₁

C₂

semidirect

60. C₈×C₁₅₁

#41

semidirect

76. C₁₅₁

#3184

semidirect

80. #493

D₁₅₁

semidirect

137. C₈

#14559003

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#23752760	2
#23752610	2
#23752558	2
#23752462	2
#23752324	2
C₄×#14558987	2
#493×C₁₅₁	2
#3184	151

subgroup

index

#493×C₁₅₁

151

Subgroups — #34079846

Conjugacy classes of subgroups

Complements

Maximal subgroups